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3 Answers
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There is no reason why you should not publish papers between postdocs. Journals do not require authors to have affiliations. However, not having an affiliation may mean that your work is given more scrutiny under peer-review.

If the work is in any way speculative or outside current areas of research, then not having an affiliation or a strong publication history may make it difficult to publish in a high impact journal. You can settle for a less prestigious journal or just submit to arXiv, assuming you keep your submission rights or the support of an endorser. Failing that, there are other archives or you can start a blog to use some of your spare time and improve your reputation.

There should not be a problem in submitting a paper for publication without a current affiliation. The problem would be at the level of how seriously it would be taken by the journal and the peer review if it is not under a the auspices of a formal institution. So collaborating with somebody in an institute may help the process of approval.

In this internet age there exist the ArXiv.org where people can submit their papers without peer review, only a cursory moderation of relevance.Wikipedia has an article on how it works. Often the preprints submitted there end up having a journal reference too, once they go through the tedious journal peer review.

Thanks for your answer. I do use arXiv, but many postdoctoral fellowship applications ask for a list of articles published in a peer review journal only - they don't seem to care about unpublished ones.
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Space boyJan 24 '12 at 13:03

If you publish work that was done during your postdoc your affiliation should be that of the postdoc. However, use a private email address (such as gmail) and as soon as you can add to the manuscript a footnote with the current address.

Even if you haven't found funding yet I see no reason that the department of your last postdoc (or of your Ph.D.) will not agree that you remain affiliated with them.

PLoS ONE may be a good choice for submitting as it "will rigorously peer-review your submissions and publish all papers that are judged to be technically sound."

If you have difficulties finding a postdoc position this may indicate that it will even be more difficult to find an academic position. This would be a good time to start considering also career paths outside of Academia.